Planar triangular $S=3/2$ magnet ${\mathrm{AgCrSe}}_2$: Magnetic frustration, short range correlations, and field-tuned anisotropic cycloidal magnetic order

Single crystals of the hexagonal triangular lattice compound ${\mathrm{AgCrSe}}_2$ have been grown by chemical vapor transport. The crystals have been carefully characterized and studied by magnetic susceptibility, magnetization, specific heat, and thermal expansion. In addition, we used Cr-electron spin resonance and neutron diffraction to probe the Cr $3d^3$ magnetism microscopically. To obtain the electronic density of states, we employed x-ray absorption and resonant photoemission spectroscopy in combination with density functional theory calculations. Our studies evidence an anisotropic magnetic order below $T_N=32\mathrm{K}$. Susceptibility data in small fields of about 1 T reveal an antiferromagnetic (AFM) type of order for $H\perp c$, whereas for $H\parallel c$ the data are reminiscent of a field-induced ferromagnetic (FM) structure. At low temperatures and for $H\perp c$, the field-dependent magnetization and AC susceptibility data evidence a metamagnetic transition at $H^{+}=5\mathrm{T}$, which is absent for $H\parallel c$. We assign this to a transition from a planar cycloidal spin structure at low fields to a planar fanlike arrangement above $H^{+}$. A fully ferromagnetically polarized state is obtained above the saturation field of $H_{\perp S}=23.7\mathrm{T}$ at 2 K with a magnetization of $M_s=2.8{\mu }_{\mathrm{B}}/\mathrm{Cr}$. For $H\parallel c$, $M\left(H\right)$ monotonically increases and saturates at the same $M_s$ value at $H_{\parallel S}=25.1\mathrm{T}$ at 4.2 K. Above $T_N$, the magnetic susceptibility and specific heat indicate signatures of two dimensional (2D) frustration related to the presence of planar ferromagnetic and antiferromagnetic exchange interactions. We found a pronounced nearly isotropic maximum in both properties at about $T^{*}=45\mathrm{K}$, which is a clear fingerprint of short range correlations and emergent spin fluctuations. Calculations based on a planar 2D Heisenberg model support our experimental findings and suggest a predominant FM exchange among nearest and AFM exchange among third-nearest neighbors. Only a minor contribution might be assigned to the antisymmetric Dzyaloshinskii-Moriya interaction possibly related to the noncentrosymmetric polar space group $R3m$. Due to these competing interactions, the magnetism in ${\mathrm{AgCrSe}}_2$, in contrast to the oxygen-based delafossites, can be tuned by relatively small, experimentally accessible magnetic fields, allowing us to establish the complete anisotropic magnetic H-T phase diagram in detail.

Figure 1

Hexagonal lattice structure of ${\mathrm{AgCrSe}}_2$ (left) and morphology of two crystals used in this study (right). The lateral extension corresponds to the $(a,b)$ plane and the direction perpendicular to it is the $c$ axis.

Figure 2

(a) X-ray absorption spectroscopy at the Cr $L_{2,3}$ edge, indicating a dominant ${\mathrm{Cr}}^{3+}$ valence. (b) Angle-integrated resonant photoemission measurements of the valence band electronic structure, measured at the photon energies marked by the points in (a) which span across the $L_3$ edge. (c) The Cr partial density of states extracted experimentally as the difference of the on- and off-resonant PES spectra ($h\nu =576.5$ eV and 569.4 eV, respectively; solid blue line with shading). The dashed lines show the corresponding Cr partial DOS as determined by nonmagnetic DFT, and by $\mathrm{LDA}+U$ calculations for $U=0.75$ eV and 3 eV, respectively.

Figure 3

Total and partial electronic densities of states of ${\mathrm{AgCrSe}}_2$. (a) Nonmagnetic metallic solution with dominating Cr $3d$ states at $E_F$. (b) Ferromagnetic insulating state including spin-orbit coupling for $U=0.75$ eV and (c) $U=3$ eV.

Figure 4

Temperature dependence of the magnetic susceptibility (${\chi }_{\parallel }$ and ${\chi }_{\perp }$) of ${\mathrm{AgCrSe}}_2$ (MS1212) measured at ${\mu }_{0}H=1\mathrm{T}$ (left). The solid line corresponds to a Curie-Weiss fit. Each measurement was first performed in zero-field-cooled (zfc) and then in field-cooled (fc) mode. Inverse of the susceptibility as a function of temperature (right). Note that a paramagnetic contribution for $H\parallel c$ of ${\chi }_{0,\parallel }=0.0005$ emu/mol was required for a proper Curie-Weiss fit (CW) at high temperatures. Most likely this stems from a Van Vleck contribution (${\chi }_{\mathrm{VV}}$) (estimated from high field measurements; Fig. 9) and a small and negative diamagnetic contribution from the capton sample holder ${\chi }_{0,\parallel }={\chi }_{\mathrm{VV},\parallel }+{\chi }_{\mathrm{dia}}$.

Figure 5

First temperature derivative of the magnetic susceptibility (${\chi }_{\parallel }$ and ${\chi }_{\perp }$) of ${\mathrm{AgCrSe}}_2$ (MS1212) as a function of temperature. The maximum in susceptibility at $T^{*}=46.3\mathrm{K}$ (Fig. 4) is given by the high temperature zero crossing of the derivative. The ordering temperature of $T_N=32\mathrm{K}$ is indicated by a maximum in the derivative for $H\perp c$ and the second zero crossing in the derivative for $H\parallel c$.

Figure 6

Temperature dependence of the magnetic susceptibility of ${\mathrm{AgCrSe}}_2$ (MS1212) measured at various magnetic fields for $H\perp c$ (left) and $H\parallel c$ (right). All measurements are performed in zfc/fc mode.

Figure 7

AC susceptibility (a) and DC magnetization (b) versus field for $H\parallel c$ and $H\perp c$ at various temperatures as indicated. The DC and AC magnetization were measured following a zfc protocol and by increasing the field from zero to 14 T followed by warming up to 200 K between the individual cycles.

Figure 8

Magnetization versus field for $H\perp c$. Open symbols represent data from measurements in static fields in a PPMS (at the MPI) and solid lines represent high field magnetization data recorded in pulsed fields at the HLD.

Figure 9

Magnetization versus field for $H\parallel c$. Open symbols represent data from measurements in static fields in a PPMS and solid lines represent high field magnetization data obtained in pulsed fields at the HLD. The dashed line corresponds to the linear in field Van Vleck contribution ($M_{\mathrm{VV}}=H{\chi }_{\mathrm{VV}\parallel }$) above 40 T with ${\chi }_{\mathrm{VV}\parallel }=0.0008$ emu/mol. The inset shows the first field derivative of the magnetization versus field for $H\parallel c$ at various temperatures. The saturation field $H_S$ is given by a step in the derivatives.

Figure 10

First field derivatives of the magnetization versus field for $H\perp c$ at various temperatures. The metamagnetic transition field $H^{+}$ is defined by a pronounced peak, whereas the onset of saturation at $H_S$ is given by a step in the derivatives.

Figure 11

(a) Temperature dependence of the zero-field specific heat $C_p$ of an ${\mathrm{AgCrSe}}_2$ single crystal. (b) $C_p/T$ vs $T$ for various fields for $H\parallel c$ (left axis) and magnetic specific heat $C_{\mathrm{mag}}$ at zero field as function of temperature (right axis). The maximum in $C_{\mathrm{mag}}$ at $T^{*}=43\mathrm{K}$ is clearly visible and the maximum value $C_{\mathrm{mag}}\left(43\mathrm{K}\right)$ corresponds to $0.36R$.

Figure 12

Thermal expansion coefficient $\alpha$ measured in $c$ direction as function of temperature (left scale) together with $C_p/T$ (expanded view) (right scale). The dashed vertical line marks the largest anomaly in the expansion at $T_N\approx 33\mathrm{K}$.

Figure 13

(a) Evolution of the ${\mathrm{Cr}}^{3+}$ ESR spectra with temperature, (b) $g$ factor anisotropy at 295 K, and temperature dependence of ESR linewidth and (c) inverse ESR intensity for magnetic fields $H\perp c$ and $H\parallel c$. Solid lines are fits to the linewidth data with a model assuming a spin relaxation via $Z_2$ vortices (see Refs. [72, 74] and main text) and a Curie-Weiss law, respectively.

Figure 14

(a) Rietveld refinement of the neutron diffraction data collected at $T=1.5\mathrm{K}$ with the forward-scattering detector bank [$R_{\mathrm{Bragg}}\left(\mathrm{nuclear}\right)=6.24\%$ and $R_{\mathrm{Bragg}}\left(\mathrm{magnetic}\right)=3.48\%$]. The cross symbols and solid red line represent the experimental and calculated intensities, respectively, and the blue line below is the difference between them. Tick marks (green) indicate the positions of Bragg peaks: nuclear (top), magnetic (middle), and vanadium can (bottom). Inset shows integrated intensity of the diffraction pattern in the $d$-spacing range of 12–15 Å, where the strongest magnetic satellite $(0.037,0.037,3/2)$ is observed, as a function of temperature. (b) Schematic representation of the long-period modulated magnetic structure of ${\mathrm{AgCrSe}}_2$.

Figure 15

Unit cell volume as a function of temperature. Inset shows the temperature dependence of the unit cell parameters.

Figure 16

H-T phase diagram of ${\mathrm{AgCrSe}}_2$, obtained from various methods for magnetic fields $H\perp c$ and $H\parallel c$. The ordered state 1 represents the cycloidal phase for $H\perp c$ below $H^{+}\left(T\right)$ [see Fig. 14], whereas phase 2 represents a planar fan phase for in-plane fields [above $H^{+}\left(T\right)$] and a more conelike phase for fields in $c$ direction.